Studies of the Social Cost of Carbon assume climate change is a stock externality for which damages stem from warming level. However, economic and natural systems are also sensitive to the rate at which warming occurs. In this paper, I study the optimal carbon tax when such a feature is accounted for. I show that damages caused by warming rates do not affect optimal long-term warming, but they delay the use of the same carbon budget. Numerically, optimal carbon price should be 39% higher, compared to when damages solely stem from warming level.